Editing Monkey saddle

{{short description|1=Mathematical surface defined by z = x³ – 3xy²}}
[[Image:Monkey Saddle Surface (Shaded).png|thumb|The monkey saddle|300px]]
In [[mathematics]], the '''monkey saddle''' is the [[surface (mathematics)|surface]] defined by the equation 
:<math> z = x^3 - 3xy^2, \, </math>
or in [[Cylindrical coordinate system|cylindrical coordinates]]
:<math>z = \rho^3 \cos(3\varphi).</math>

It belongs to the class of [[saddle surface]]s, and its name derives from the observation that a [[saddle]] used by a [[monkey]] would require two depressions for its legs and one for its tail. The point {{tmath|(0,0,0)}} on the monkey saddle corresponds to a [[critical point (mathematics)|degenerate critical point]] of the function {{tmath|z(x,y)}} at {{tmath|(0, 0)}}. The monkey saddle has an isolated [[umbilical point]] with zero [[Gaussian curvature]] at the origin, while the curvature is strictly negative at all other points.

One can relate the rectangular and cylindrical equations using [[complex number]]s <math>x+iy = r e^{i\varphi}:</math>

:<math> z = x^3 - 3xy^2 = \operatorname{Re} [(x+iy)^3] = \operatorname{Re}[r^3 e^{3i\varphi}] = r^3\cos(3\varphi).</math>

By replacing 3 in the cylindrical equation with any integer {{tmath|k \geq 1,}} one can create a saddle with {{tmath|k}} depressions.
<ref>Peckham, S.D. (2011) Monkey, starfish and octopus saddles, ''Proceedings of Geomorphometry 2011'', Redlands, CA, pp. 31-34, https://www.researchgate.net/publication/256808897_Monkey_Starfish_and_Octopus_Saddles</ref>

Another orientation of the monkey saddle is the ''Smelt petal'' defined by <math>x+y+z+xyz=0,</math> so that the ''z-''axis of the monkey saddle corresponds to the direction {{tmath|(1,1,1)}} in the Smelt petal.<ref>{{Cite book|last=J.|first=Rimrott, F. P.|title=Introductory Attitude Dynamics|date=1989|publisher=Springer New York|isbn=9781461235026|location=New York, NY|pages=26|oclc=852789976}}</ref><ref>{{Cite journal|last=Chesser|first=H.|last2=Rimrott|first2=F.P.J.|date=1985|editor-last=Rasmussen|editor-first=H.|title=Magnus Triangle and Smelt Petal|journal=CANCAM '85: Proceedings, Tenth Canadian Congress of Applied Mechanics, June 2-7, 1985, the University of Western Ontario, London, Ontario, Canada}}</ref>

Another function, which has not three but four areas - in each quadrant of the <math>\mathbb R^2</math>, in which the function goes to minus infinity, is given by <math>z = x^4 - 6x^2y^2 + y^4</math>.

[[File:Shape_petal.svg|alt=Shape petal|thumb|300x300px|Smelt petal: {{math|1=''x'' + ''y'' + ''z'' + ''xyz'' = 0}}]]

== Horse saddle ==

The term ''horse saddle'' may be used in contrast to monkey saddle, to designate an ordinary saddle surface in which ''z''(''x'',''y'') has a [[saddle point]], a local minimum or maximum in every direction of the ''xy''-plane. In contrast, the monkey saddle has a stationary [[point of inflection]] in every direction.

==References==
<references />

==External links==
* {{MathWorld | urlname=MonkeySaddle | title=Monkey Saddle}}

[[Category:Multivariable calculus]]
[[Category:Algebraic surfaces]]